Expand

P_{N} w

P_{N} w = \sum_{k = 1}^{N} w_{k} ϕ_{k} .

The

H

and

V

norms satisfy

∣ P_{N} w ∣^{2} = \sum_{k = 1}^{N} ∣ w_{k} ∣^{2}, ∥ P_{N} w ∥^{2} = (A P_{N} w, P_{N} w) = \sum_{k = 1}^{N} λ_{k} ∣ w_{k} ∣^{2} .

Since

λ_{k} \leq λ_{N}

on the projected space,

∥ P_{N} w ∥ \leq λ_{N}^{1/2} ∣ P_{N} w ∣ .

ii

Solution

L^{2}

and

L^{6}

, followed by the three-dimensional Sobolev inequality, gives

∥ v ∥_{L^{3}} \leq ∥ v ∥_{L^{2}}^{1/2} ∥ v ∥_{L^{6}}^{1/2} \leq c ∣ v ∣^{1/2} ∥ v ∥^{1/2} .

Apply this with

v = P_{N} w

and use part i:

∥ P_{N} w ∥_{L^{3}} \leq c ∣ P_{N} w ∣^{1/2} (λ_{N}^{1/2} ∣ P_{N} w ∣)^{1/2} = c λ_{N}^{1/4} ∣ P_{N} w ∣.

H

, so

∥ P_{N} w ∥_{L^{3}} \leq c λ_{N}^{1/4} ∣ w ∣ .

The Sobolev constant is dimensionless after using the periodic-domain normalization, so the estimate is scale invariant.

By the Holder inequality with exponents

3, 2, 6

∣ ⟨ B (P_{N} u, v), w ⟩ ∣ = \int_{Ω} ((P_{N} u) \cdot \nabla v) \cdot w d x \leq ∥ P_{N} u ∥_{L^{3}} ∥\nabla v ∥_{L^{2}} ∥ w ∥_{L^{6}} .

Part ii and the Sobolev embedding

H^{1} ↪ L^{6}

yield

∣ ⟨ B (P_{N} u, v), w ⟩ ∣ \leq c λ_{N}^{1/4} ∣ u ∣ ∥ v ∥ ∥ w ∥ .

Equivalently,

∥ B (P_{N} u, v) ∥_{V^{'}} \leq c λ_{N}^{1/4} ∣ u ∣ ∥ v ∥.

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